Question

\(P\) is a point on the circumference of a circle \(X\) of diameter \(\quantity{3}{in.}\) An arc of another circle \(Y\) is drawn with centre \(P\), intersecting \(X\) at \(Q\) and \(R\). Express the lengths of \(PQ\) and the minor arc \(QR\) of \(Y\) in terms of \(\theta\), where \(\theta\) is the angle, in radians, between \(PQ\) and the diameter of \(X\) through \(P\).

Prove that the length of the arc \(QR\) has a stationary value when \(\theta = \cot \theta\), and that this value is a maximum.